The component-count conjecture for cubic Poincare systems

Let $X_{\infty} \subset \mathbb{A}^{14}$ be the algebraic set of inhomogeneous degree $3$ Poincaré differential equations over $\mathbb{C}$ for which all focal values vanish. The codimension of a component is its codimension in $\mathbb{A}^{14}$.

Component-count conjecture. The variety $X_{\infty}$ has exactly $1$ component of codimension $5$, exactly $2$ components of codimension $6$, exactly $4$ components of codimension $7$, at least one component of codimension $8$, and an unknown number of components of codimension at least $9$.

The statement records the authors' experimental estimate of the irreducible-component structure of the cubic center variety. The numbers in codimensions $5$, $6$, and $7$ are presented as recovered results, while the higher-codimension counts remain unknown in the supplied text.